4,716 research outputs found
Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach
Pairwise comparisons are a well-known method for modelling of the subjective
preferences of a decision maker. A popular implementation of the method is
based on solving an eigenvalue problem for M - the matrix of pairwise
comparisons. This does not take into account the actual values of preference.
The Heuristic Rating Estimation (HRE) approach is a modification of this method
in which allows modelling of the reference values. To determine the relative
order of preferences is to solve a certain linear equation system defined by
the matrix A and the constant term vector b (both derived from M). The article
explores the properties of these equation systems. In particular, it is proven
that for some small data inconsistency the A matrix is an M-matrix, hence the
equation proposed by the HRE approach has a unique strictly positive solution.Comment: 8 page
Heuristic Rating Estimation Method for the incomplete pairwise comparisons matrices
The Heuristic Rating Estimation Method enables decision-makers to decide
based on existing ranking data and expert comparisons. In this approach, the
ranking values of selected alternatives are known in advance, while these
values have to be calculated for the remaining ones. Their calculation can be
performed using either an additive or a multiplicative method. Both methods
assumed that the pairwise comparison sets involved in the computation were
complete. In this paper, we show how these algorithms can be extended so that
the experts do not need to compare all alternatives pairwise. Thanks to the
shortening of the work of experts, the presented, improved methods will reduce
the costs of the decision-making procedure and facilitate and shorten the stage
of collecting decision-making data.Comment: 13 page
On Axiomatization of Inconsistency Indicators for Pairwise Comparisons
We examine the notion of inconsistency in pairwise comparisons and propose an
axiomatization which is independent of any method of approximation or the
inconsistency indicator definition (e.g., Analytic Hierarchy Process, AHP). It
has been proven that the eigenvalue-based inconsistency (proposed as a part of
AHP) is incorrect.Comment: Enhanced text, with 21 pages and 3 figures, proves that arbitrarily
inaccurate pairwise matrices are considered acceptable by theories with a
inconsistency based on the principal eigenvalue (e.g., AHP). CPC (corner
pairwise comparisons) matrix is the crucial part of this study as it
invalidates any eigenvalue-based inconsistency. All comments are highly
appreciate
On the geometric mean method for incomplete pairwise comparisons
When creating the ranking based on the pairwise comparisons very often, we
face difficulties in completing all the results of direct comparisons. In this
case, the solution is to use the ranking method based on the incomplete PC
matrix. The article presents the extension of the well known geometric mean
method for incomplete PC matrices. The description of the methods is
accompanied by theoretical considerations showing the existence of the solution
and the optimality of the proposed approach.Comment: 15 page
Clustering and Inference From Pairwise Comparisons
Given a set of pairwise comparisons, the classical ranking problem computes a
single ranking that best represents the preferences of all users. In this
paper, we study the problem of inferring individual preferences, arising in the
context of making personalized recommendations. In particular, we assume that
there are users of types; users of the same type provide similar
pairwise comparisons for items according to the Bradley-Terry model. We
propose an efficient algorithm that accurately estimates the individual
preferences for almost all users, if there are
pairwise comparisons per type, which is near optimal in sample complexity when
only grows logarithmically with or . Our algorithm has three steps:
first, for each user, compute the \emph{net-win} vector which is a projection
of its -dimensional vector of pairwise comparisons onto an
-dimensional linear subspace; second, cluster the users based on the net-win
vectors; third, estimate a single preference for each cluster separately. The
net-win vectors are much less noisy than the high dimensional vectors of
pairwise comparisons and clustering is more accurate after the projection as
confirmed by numerical experiments. Moreover, we show that, when a cluster is
only approximately correct, the maximum likelihood estimation for the
Bradley-Terry model is still close to the true preference.Comment: Corrected typos in the abstrac
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